Sparseness bounds on local operators in holographic $CF{T}_d$

Abstract

We use the thermodynamics of anti-de Sitter gravity to derive sparseness bounds on the spectrum of local operators in holographic conformal field theories. The simplest such bound is $\rho (\Delta )\lesssim \exp \left(\frac{2\pi \Delta }{d-1}\right)$ for CFT${}_d$. Unlike the case of $d=2$, this bound is strong enough to rule out weakly coupled holographic theories. We generalize the bound to include spins $J_i$ and $U(1)$ charge $Q$, obtaining bounds on $\rho (\Delta ,J_i,Q)$ in $d=3$ through $6$. All bounds are saturated by black holes at the Hawking-Page transition and vanish beyond the corresponding BPS bound.

21 pages, 9 figures, 2 appendices

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• Conformal field theories dual to quantum gravity with strongly coupled matter